Chapter 3: Descriptive Measures

Frog Thumb Length. W. Duellman and J. Kohler explore a new species of frog in the article "New Species of Marsupial Frog (Hylidae: Hemiphractinae: Gastrotheca) from the Yungas of Bolivia" (Journal of Herpetology, Vol. $39$, No. $1$. pp. $91-100$). These two museum researchers collected information on the lengths and widths of different body parts for the male and female Gastrotheca piper ata. Thumb length for the female Gastrotheca piperata has a mean of $6.71$mm and a standard deviation of $0.67$mm. Let x denote thumb length for a female specimen.

a. Find the standardized version of x.

b. Determine and interpret the $2$-scores for thumb lengths of $5.2$mm and $8.1$mm. Round your answers to two decimal places.

Data on low-birth-weight babies were collected over a 2-year period by 14 participating centers of the National Institute of Child Health and Human Development Neonatal Research Network. Results were reported by J. Lemons et al. in the on-line paper "Very Low Birth Weight Outcomes of the National Institute of Child Health and Human Development Neonatal Research Network". For the 1048 surviving babies whose birth weights were 751-1000 grams, the average length of stay in the hospital was 86 days, although one center had an average of 66 days and another had an average of 108 days.

Part (a): Can the mean lengths of stay be considered population means? Explain your answers.

Part (b): Assuming that the population standard deviation is 12 days, determine the z-score for a baby's length of stay of 86 days at the center where the mean was 66days.

Part (c): Assuming that the population standard deviation is 12 days, determine the z-score for a baby's length of stay of 86 days at the center where the mean was 108days.

Part (d): What could you conclude from parts (b) and (c) about an infant with a length of stay equal to the mean at all centers if that infant was born at a center with a mean of 66 days? mean of 108days?

Suppose you buy a new car whose advertised mileage is 25 miles per gallon (mpg). After driving your car for several months, you find that its mileage is 21.4mpg. You telephone the manufacturer and learn that the standard deviation of gas mileages for all cars of the model you bought is 1.15 mpg.

Part (a): Find the z-score for the gas mileage of your car, assuming the advertised claim is correct.

Part (b): Does it appear that your car is getting unusually low gas mileage?

Suppose that you take an exam with 400 possible points and are told that the mean score is 280 and that the standard deviation is 20. you are also told that you got 350. Did you do well on the exam? Explain you answer.

Consider the following three data sets.

Part (a): Assuming that each of these data sets is sample data, compute the standard deviations. (Round the answers to two decimals places)

Part (b): Assuming that each of these data sets is population data, compute the standard deviations. (Round the answers to two decimals places)

Part (c): Using your results from parts (a) and (b), make an educated guess about the answer to the following question: If both s and $\sigma$are computed for the same data set, will they tend to be closer together if the data set is large or if it is small?

Find the

a. Mean b. median c. mode

For the mean and the median, round each answer to one more decimal place than that used for the observations.

Technical Merit.In one Winter Olympics, Michelle Kwan competed in the Short Program ladies singles event. From nine judges, she received scores ranging from I (poor) to 6 (perfect). The following table provides the scores that the judges gave her on technical merit, found in an article by S. Berry .

$5.85.75.95.75.55.75.75.75.6$

Consider a data set with m observations. If the data are sample data, you compute the sample standard deviation, s, whereas if the data are population data, you compute the population standard deviation, $\sigma$.

Part (a): Derive a mathematical formula that gives $\sigma$ in terms of s when both are computed for the same data set.

Part (b): Refer to the three data sets in Exercise 3.219. Verify that your formula in part (a) works for each of the three data sets.

Part (c): Suppose that a data set consists of 15 observations. You compute the sample standard deviation of the data and obtain $s=38.6$. Then you realize that the data are actually population data and that you should have obtained the population standard deviation instead. Use your formula from part (a) to obtain$\sigma$.

Dallas Mavericks. From the ESPN website, in the Dallas Mavericks Roster, we obtained the following weights, in pounds, for the players on that basketball team for the $2013-2014$season.

Use the technology of your choice to determine

a. the population mean weight.

b. the population standard deviation of the weights. Note: Depending on the technology that you're using, you may need to refer to the formula derived in Exercise$3.220\left(a\right)$.

A company produces cans of stewed tomatoes with an advertised weight of 14oz. The standard deviation of the weights is known to be 0.4oz. A quality - control engineer selects a can of stewed tomatoes at random and finds its net weight to be 17.28oz.

Part (a): Estimate the relative standing of that can of stewed tomatoes, assuming the true mean weight is 14oz. Use the z-score and Chebyshev's rule.

Part (b): Does the quality - control engineer have reason to suspect that the true mean weight of all cans of stewed tomatoes being produced is not 14oz? Explain your answer.

Suppose that you are thinking of buying a resale home in a large tract. The owner is asking \(205,500. Your realtor obtains the sale prices of comparable homes in the area that have sold recently. The mean of the prices is \)220,258 and the standard deviation is $5,237. Does it appear that the home you are contemplating buying is a bargain? Explain your answer using the z-score and Chebyshev's rule.